Find eqation for:$\ A^n$ when:
$$A =\left( \begin{array}{cc} a & 1 & 0\\ 0 & a & 1\\ 0 & 0 & a\end{array} \right)$$
I calculated $\ A^2$ $\ A^3$ and compared entries:
At the end I've got something like:
$$A^n =\left( \begin{array}{cc} a^n & na^{n-1} & ???\\ 0 & a^n & na^{n-1}\\ 0 & 0 & a^n\end{array} \right)$$
So I'm struggling with estimating value of the last entry.
From my calculations I've got respectively: $\ 0 $ $\ 1 $ $\ 3a $. How I can find the equation for this sequence?
On
Hint1: Find the eigenvalues and eigenvectors and diagonalize the matrix
$A = P^{-1}BP$ then $A^n = P^{-1}B^nP$.
Here $B$ is a diagonal matrix so $n$th power of $B$ will be easy to find.
Hint2: You can also find nth power of $A$ by using char polynomial.
On
$$A^2 = \left( \begin{array}{ccc} a^2 & 2 a & 1 \\ 0 & a^2 & 2 a \\ 0 & 0 & a^2 \\ \end{array} \right)$$
$$A^3 = \left( \begin{array}{ccc} a^3 & 3 a^2 & 3 a \\ 0 & a^3 & 3 a^2 \\ 0 & 0 & a^3 \\ \end{array} \right)$$
$$A^4 = \left( \begin{array}{ccc} a^4 & 4 a^3 & 6 a^2 \\ 0 & a^4 & 4 a^3 \\ 0 & 0 & a^4 \\ \end{array} \right)$$
$$A^5 = \left( \begin{array}{ccc} a^5 & 5 a^4 & 10 a^3 \\ 0 & a^5 & 5 a^4 \\ 0 & 0 & a^5 \\ \end{array} \right)$$
Now the sequence in the top right element:
$$1, 3, 6, 10, ...$$
Doesn't remember you about the binomial expansion? :)
Hint : let $B=\begin{pmatrix} 0&1&0\\0&0&1\\0&0&0 \end{pmatrix}$. You have $A=aI+B$. Then use the binomial theorem.